Mathematical Treasures - Gerolamo Cardano's Practica Arithmetice

This is the title page of the Practica Arithmeticeof Gerolamo Cardano (1501-1576), published in 1539. It was a comprehensive work on arithmetical questions, with numerous practical problems and even some elementary algebra and geometry.

In paragraph 12, Cardano details the side length of regular polygons inscribed in a circle of diameter 10000. Thus the side of an equilateral triangle is 8860, of a square is 7071, and of a hexagon is 5000.

In paragraph 63, Cardano solves a classical problem coming from Leonardo of Pisa: To find a number that when divided by each of 2, 3, 4, 5, and 6 falls short by 1 of exactly dividing, but is divisible by 7. The answer is 119.

In paragraph 97, Cardano solves another problem coming from Leonardo: If of four men, the first, second, and third together have 34 coins; the first, second, and fourth have 73; the first, third, and fourth have 72; and the second, third, and fourth have 88; then how many coins does each have? Cardano simply adds the four numbers together, noting that this sum, 267, is equal to three times the total. Therefore, the total number of coins is 89 and it is straightforward to calculate the number held by each man.

In paragraph 122, Cardano solves a cubic equation without using any formula.